Is nihilism the logical extreme of atheism?

[Angrboda]

I'm not an atheist, but my belief is that the common understanding of the nature of meaning is an illusion, an illusion consisting of material processes.

But aren't illusions, of necessity, in some way meaningful?

Not in the sense you are trying to indicate. Our brains appear capable of interpreting anything as meaningful, regardless of whether the sense of our thoughts represent anything or not. Experiments with drugs like DMT and hallucinations make clear that the brain is quite capable of declaring anything as "seemingly meaningful." Meaningfulness appears to be a natural category of our minds, like the categories moral and immoral, or the belief in free choice; whether there is an objective construct behind these categories which corresponds with their appearance is as yet unclear. (As is often the case, senselessness can easily go unnoticed, such as times when you fail to question what you've just heard or read. It can also arbitrarily appear where it shouldn't, such as in the case of jamais vu. Or as in the case with some aphasiacs who are unaware that their vocal productions don't make sense, a mix of aphasia and anosognosia.)